Optimal. Leaf size=45 \[ -\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3}{24 c^2}+\frac {d^2 (b+2 c x)^5}{40 c^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {697}
\begin {gather*} \frac {d^2 (b+2 c x)^5}{40 c^2}-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^3}{24 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int (b d+2 c d x)^2 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^2}{4 c}+\frac {(b d+2 c d x)^4}{4 c d^2}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3}{24 c^2}+\frac {d^2 (b+2 c x)^5}{40 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 64, normalized size = 1.42 \begin {gather*} d^2 \left (a b^2 x+\frac {1}{2} b \left (b^2+4 a c\right ) x^2+\frac {1}{3} c \left (5 b^2+4 a c\right ) x^3+2 b c^2 x^4+\frac {4 c^3 x^5}{5}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 79, normalized size = 1.76
method | result | size |
gosper | \(\frac {x \left (24 c^{3} x^{4}+60 x^{3} b \,c^{2}+40 x^{2} c^{2} a +50 b^{2} c \,x^{2}+60 a b c x +15 b^{3} x +30 a \,b^{2}\right ) d^{2}}{30}\) | \(61\) |
norman | \(\left (\frac {4}{3} d^{2} c^{2} a +\frac {5}{3} b^{2} c \,d^{2}\right ) x^{3}+\left (2 a b c \,d^{2}+\frac {1}{2} b^{3} d^{2}\right ) x^{2}+a \,b^{2} d^{2} x +\frac {4 c^{3} d^{2} x^{5}}{5}+2 b \,c^{2} d^{2} x^{4}\) | \(78\) |
default | \(\frac {4 c^{3} d^{2} x^{5}}{5}+2 b \,c^{2} d^{2} x^{4}+\frac {\left (4 d^{2} c^{2} a +5 b^{2} c \,d^{2}\right ) x^{3}}{3}+\frac {\left (4 a b c \,d^{2}+b^{3} d^{2}\right ) x^{2}}{2}+a \,b^{2} d^{2} x\) | \(79\) |
risch | \(\frac {4}{5} c^{3} d^{2} x^{5}+2 b \,c^{2} d^{2} x^{4}+\frac {4}{3} d^{2} x^{3} c^{2} a +\frac {5}{3} d^{2} b^{2} c \,x^{3}+2 d^{2} a b c \,x^{2}+\frac {1}{2} d^{2} b^{3} x^{2}+a \,b^{2} d^{2} x\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 71, normalized size = 1.58 \begin {gather*} \frac {4}{5} \, c^{3} d^{2} x^{5} + 2 \, b c^{2} d^{2} x^{4} + a b^{2} d^{2} x + \frac {1}{3} \, {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (b^{3} + 4 \, a b c\right )} d^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.53, size = 71, normalized size = 1.58 \begin {gather*} \frac {4}{5} \, c^{3} d^{2} x^{5} + 2 \, b c^{2} d^{2} x^{4} + a b^{2} d^{2} x + \frac {1}{3} \, {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (b^{3} + 4 \, a b c\right )} d^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (41) = 82\).
time = 0.01, size = 85, normalized size = 1.89 \begin {gather*} a b^{2} d^{2} x + 2 b c^{2} d^{2} x^{4} + \frac {4 c^{3} d^{2} x^{5}}{5} + x^{3} \cdot \left (\frac {4 a c^{2} d^{2}}{3} + \frac {5 b^{2} c d^{2}}{3}\right ) + x^{2} \cdot \left (2 a b c d^{2} + \frac {b^{3} d^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.02, size = 79, normalized size = 1.76 \begin {gather*} \frac {4}{5} \, c^{3} d^{2} x^{5} + 2 \, b c^{2} d^{2} x^{4} + \frac {5}{3} \, b^{2} c d^{2} x^{3} + \frac {4}{3} \, a c^{2} d^{2} x^{3} + \frac {1}{2} \, b^{3} d^{2} x^{2} + 2 \, a b c d^{2} x^{2} + a b^{2} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 69, normalized size = 1.53 \begin {gather*} \frac {4\,c^3\,d^2\,x^5}{5}+\frac {c\,d^2\,x^3\,\left (5\,b^2+4\,a\,c\right )}{3}+2\,b\,c^2\,d^2\,x^4+\frac {b\,d^2\,x^2\,\left (b^2+4\,a\,c\right )}{2}+a\,b^2\,d^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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